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Use the Laplace transform to solve the following initial value problem: 1/' - 47 + 4y = 0 3/(0) = -2, 1/(0) = 3 a. Using Y for the Laplace transform of y(t), i.e., Y = [(y(t) ). find the equation you get by taking the Laplace transform of the differential equation . Solve for Y(s) = c. write the above answer in its partial fraction decomposition, Y(s) = mia + Today Y(8) =+ d . Now , by inverting the transform, find y(t) = Use the Laplace transform to solve the following initial value problem: y' - 4y + 20y=0 y(0) = 0, y(0) =4 a. Using Y for the Laplace transform of y(t), i.e., Y = Lty(t)}, find the equation you get by taking the Laplace transform of the differential equation =0 b. Now solve for Y(s ) = c. By completing the square in the denominator and inverting the transform, find y(t ) =- Consider the initial value problem y' + 16y = 64t, y(0) =4, y(0) = 9. a. Take the Laplace transform of both sides of the given differential equation to create the corresponding algebraic equation. Denote the Laplace tran y(t) by Y(s). Do not move any terms from one side of the equation to the other (until you get to part (b) below). 0=[help (formulas) b. Solve your equation for Y(s). Y(s) = [ty(t)) = c. Take the inverse Laplace transform of both sides of the previous equation to solve for y(t). y(t) = Use the Laplace transform to solve the following initial value problem: y" + 9y = cos(8t) y(0) = 0, 3/(0) = 0 First, using Y for the Laplace transform of y(t), i.e., Y = [ty(t) }. find the equation you get by taking the Laplace transform of the differential equation and solving for Y: Y(s ) = Find the partial fraction decomposition of Y(s) and its inverse Laplace transform to find the solution of the IVP: y (t )